Algebraic Fractions (Edexcel GCSE Maths: Higher): Exam Questions

Simplify fully     $\frac{2x^2-5x+3}{x^2+5x-6}$ 

Write as a single fraction in its simplest form   $\frac{5}{2-x}-\frac{4}{x}$

Simplify  $\frac{x^2-16}{2x^2-5x-12}$

Show that $\frac{x^2+5x+6}{3x^2+7x-6}$ can be written in the form $\frac{ax+b}{cx+d}$ where $a$, $b$, $c$ and $d$ are integers.

Simplify fully  $\frac{3x^2-8x-3}{2x^2-6x}$

Show that  $\frac{2x^2-3x-5}{x^2+6x+5}$  can be written in the form  $\frac{ax+b}{cx+d}$ where $a,b,c$ and $d$ are integers.

Simplify fully    $\frac{x^2+3x-4}{2x^2-5x+3}$

Write $\frac{4}{x+2}+\frac{3}{x-2}$  as a single fraction in its simplest form.

Show that  $\frac{a}{b+1}-\frac{a}{{\left(b+1\right)}^2}$ can be written as  $\frac{ab}{{\left(b+1\right)}^2}$

Simplify      $\frac{x^2-9}{2x^2+5x-3}$

Solve $\frac{4-3x}{5}-\frac{3x-5}{2}=-3$

Show clear algebraic working.

$x$ = .......................

Solve the equation

$\frac{5}{x+2}+\frac{3}{x^2+2x}=2$

Show clear algebraic working.

Simplify fully $\frac{x^5-4x^3}{3x-6}$

Show that, for $x\ne –1$

   $\frac{8x^2-8}{4x+4}$ simplifies to the form $ax+b$       
where $a$ and $b$ are integers.

Show that $\frac{x+9}{x^2-1}+\frac{4}{x+1}$ can be written in the form $\frac{a}{x-1}$ where $a$ is an integer.

Show that $\frac{5x}{x+5}+\frac{25}{x-7}-\frac{300}{\left(x+5\right)\left(x-7\right)}$ simplifies to an integer.

$2-\frac{x+2}{x-3}-\frac{x-6}{x+3}$  can be written as a single fraction in the form $\frac{ax+b}{x^2-9}$

where $a$and $b$ are integers.

Work out the value of $a$ and the value of $b$.

Show that $\frac{1}{6x^2+7x-5}÷\frac{1}{4x^2-1}$ simplifies to $\frac{ax+b}{cx+d}$ where $a,b,c$ and $d$ are integers.

Simplify fully       $\frac{3-x}{3x^2-5x-12}$

Write $\frac{x}{x-1}-\frac{x}{x+1}$ as a single fraction in its simplest form.

Write

      $4-\left[\left(x+3\right)÷\frac{x^2+5x+6}{x-2}\right]$

as a single fraction in its simplest form.
You must show your working.

Show that  $\frac{3x+6}{x^2-3x-10}÷\frac{x+5}{x^3-25x}$ simplifies to $ax$ where $a$ is an integer.

Show that $6+\left[\left(x+5\right)÷\frac{x^2+3x-10}{x-1}\right]$  simplifies to $\frac{ax-b}{cx-d}$  where $a,b,c$ and $d$ are integers.

Write $\frac{25x^2-64}{5x^2-13x-6}\times \frac{x^2-8x+15}{5x+8}-\left(x-7\right)$

as a single fraction in its simplest form.
Show clear algebraic working.

Given that  $x=\frac{5}{9y+5}$ and that $y=\frac{5}{5a-2}$

find an expression for $x$ in terms of $a$.
Give your expression as a single fraction in its simplest form.

Express  $\frac{1}{9x^2-25}-\frac{1}{6x+10}$ as a single fraction in its simplest form.

Simplify  $\frac{2^n-1}{4^n-1}$

The flight of a plane was in two stages.

The table shows information about the flight.

In total, the flight lasted 2 hours.
Work out the value of $x$.

Solve  $\frac{5}{4x+1}=\frac{2x}{x^2+3}$

Give your solutions to 3 significant figures.
You must show your working.

Solve  $\frac{x}{x+4}+\frac{7}{x-2}=1$

You must show your working.

$x$=.....................

Solve $\frac{x}{4}-\frac{2x}{x+2}=1$

Give your solutions to 2 decimal places.
You must show your working.

Solve this equation, giving your answers correct to 1 decimal place.

$\frac{5}{x+2}+\frac{3}{x-3}=2$

$x$ = ........... or $x$ = ...........